TRANSIENT DISTURBANCES IN A RELAXING THERMOELASTIC HALF SPACE DUE TO MOVING INTERNAL HEAT SOURCE

This paper is concerned with the transient waves created by a line heat source that suddenly starts moving with a uniform velocity inside a thermoelastic semi-infinite medium with thermal relaxation ofthe type ofLord and Shulman The source moves parallel to the boundary surface which is traction-free. The problem is reduced to the solution of three differential equations, one involving the elastic vector potential, and the other two coupled, involving the thermoelastic scalar potential and the temperature. Using Fourier and Laplace transforms, the solution for the displacements have been obtained in the transform domain. The displacements have been calculated on the boundary surface for small time


INTRODUCTION
The problem of heat sources acting in an elastic body is one of mathematical interest as well as of physical importance The dynamic heat source problem was first investigated by Danilovskaya 1] using the uncoupled theory of thermoelasticity.The problem of instantaneous and moving heat sources in infinite and semi-infinite space, and static line heat sources in semi-infinite space were considered by Eason and Sneedon [2], Nowacki [3] and others, under the coupled theory of thermoelasticity Dhaliwal and Singh [4] gave the short time approximation due to a suddenly applied point source inside an infinite space.Nariboli and Nyayadhish [5] gave exact solutions of the one-dimensional coupled problem of impulse and thermal shock at the end of a semi-infinite rod for small time.
However, the coupled theory of thermoelasticity suffers from a serious drawback, namely, the heat conduction equation is parabolic and consequently predicts an infinite velocity for heat propagation To remedy this defect the theory of generalized thermoelasticity with one relaxation time was formulated by Lord and Shulman [6].The heat conduction equation here is a hyperbolic one so that there is a finite speed of propagation for thermal waves.
Nayfeh and Nemat Nasser [7,8] studied the problem of transient waves in thermally relaxing solids Wang and Dhaliwal [9] have studied the fundamental solutions for generalized thermoelasticity, including problems ofbody force and heat source.
In this paper, we make a study of the thermoelastic disturbances created by an internal line heat source that suddenly starts moving uniformly inside a semi-infinite space with thermal relaxation.The problem is solved by using joint Fourier and Laplace transforms.The expressions for displacements in the transform domain indicate the existence of dilatational, transverse and thermal waves inside the medium.The displacement components have been evaluated on the boundary for small time only, the general inversion being too complicated.Presence of high thermal damping makes the short time solution meaningful.

FORMULATION OF PROBLEM
We consider a homogeneous isotropic thermoelastic solid occupying the region x2 _> h which is initially at rest, and the free surface z2 h which is stress free.A line source starts moving suddenly inside the medium at a depth h below the free surface uniformly in the xl direction.The line source is parallel to the zs axis so that all quantities are independent of xs, and the third component us of the displacement vector vanishes.The governing equations are: 1. Strain displacement relation 2e, u2j +us,,, i,j 1,2. (2.1) ( Initial conditions and boundary conditions The initial conditions are u2=O, 0=0, at <0 in x2E-h ) /2=0, =0, at _<0 in x2_)-h, 1, (2.5) The stress-free boundary conditions are "r12--'r'22=0 on z2= -h for (2.6) The regularity conditions are O,ti--*O as 2--*oo, 1 --* q-O0. (2.7) The thermal boundary condition at z2 h is HIO + H20.2 =0. (2.8) Scalar and vector potentials b, (0, 0, P) are introduced as follows: where b(x,x2,) and # p(x,x2,t).
Taking Laplace and Fourier transform ofboth sides of equations (2.15)-(2.17),we have Inverting the transforms gives v -= + e a-v is the coupling parameter.and Henceforth we shall omit the bars in equation (2.15)-(2.17).We shall also write x, y for xl, x2 and u, v for ua, u2 respectively.

SMALL TIME APPROXIMATION
To the first order of approximation in 1, a and a may be written as p p(1 + p'r)  For a short time approximation to the displacement components, we expand al, 0,2, bl in terms of powers ofp, and consider relevant terms as It is clear from (4.4)-(4.6)that there are three waves with velocities B/m, 1/m2v/ and 1 respectively represeming the dilatational, the thermal and the transverse elastic waves.
However, the inversion of u*, v* inside the medium is too complicated, we evaluate the surface displacements only from (3.29) and (3.30) for the zero temperature on the boundary.
2,Qo(l + Fr)(e -'a e-'a)  vl,=_h ,. -exp{ (eh/2B3r'2) }f_ -e (I--)f4, (4 11)   where (4.12) 1 ( x mlh) ( m,h x) (413) 12=V t+r-t V 3 H 3 V It is observed that the surface displacements for small time consist of dilatational waves propagating with velocity (B/ml) and a thermal wave moving with velocity (1/m2v).Also the waves are attenuated by exponential factors depending on and r.The terms containing fl, f3 in u represent the displacement at the point x 0, while terms comaining f2, $4 represent the surface disturbance up to the point above the position of source at the time.The non-relaxing thermoelastic case may be obtained simply by putting -0 in the above results.